Advertisement

Journal of Geographical Systems

, Volume 9, Issue 3, pp 267–288 | Cite as

Integrating the maximum capture problem into a GIS framework

  • Benjamin D. Spaulding
  • Robert G. CromleyEmail author
Original Article

Abstract

This paper presents a methodology for reformulating the maximal capture problem by using the data representation and manipulation capabilities of GIS to define: (1) the coverage region captured by each potential facility, and (2) each unique demand region covered by a specific combination of potential facilities. The formulation is modeled on the maximum covering problem although the integer restriction on the demand capture variables is relaxed. Because demand regions are not exogenously given, areal interpolation is used to estimate the demand associated with each of these regions The model is used to determine the location on a network for a set of home improvement stores that are hypothetically in competition with existing Home Depot stores in Southeastern New Hampshire.

Keywords

GIS Location–allocation modeling Maximum capture problem 

JEL Classification

C6 

References

  1. Armstrong M, Densham P (1990) Database organization strategies for spatial decision support systems. Int J Geogr Inf Syst 4:3–20CrossRefGoogle Scholar
  2. Benati S (1999) The maximum capture problem with heterogeneous customers. Comput Oper Res 26:1351–1367CrossRefGoogle Scholar
  3. Benati S, Hansen P (2002) The maximum capture problem with random utilities: problem formulation and algorithms. Eur J Oper Res 143:518–530CrossRefGoogle Scholar
  4. Bunge W (1962) Theoretical geography. The Royal University of Lund, LundGoogle Scholar
  5. Casillas PA (1987) Data aggregation and the p-median problem in continuous space. In: Ghosh A, Rushton G (eds) Spatial analysis and location allocation models. Van Nostrand Reinhold Publishers, New York, pp 327–344Google Scholar
  6. Church R (1984) The planar maximal covering location problem. J Reg Sci 24:185–201CrossRefGoogle Scholar
  7. Church R (2002) Geographical information systems and location science. Comput Oper Res 29:541–562CrossRefGoogle Scholar
  8. Church R, ReVelle C (1974) The maximal covering location problem. Pap Reg Sci Assoc 32:101–118CrossRefGoogle Scholar
  9. Church R, ReVelle C (1976) Theoretical and computational links between the p-median, location set-covering, and the maximal covering location problem. Geogr Anal 8:406–415CrossRefGoogle Scholar
  10. Colomé R, Lourençe HR, Serra D (2003) A new chance-constrained maximum capture problem. Ann Oper Res 122:121–139CrossRefGoogle Scholar
  11. Cooper L (1963) Location–allocation problems. Oper Res 11:331–343Google Scholar
  12. Couclelis H (1992) People manipulate objects (but cultivate fields): beyond the raster-vector debate in GIS. In: Frank A, Campari I, Formentini U (eds) Theories and methods of spatio-temporal reasoning in geographic space. Springer, Berlin, pp 65–77Google Scholar
  13. Current JR, Schilling DA (1987) Elimination of source A and B errors in the p-median location problems. Geogr Anal 19:95–110CrossRefGoogle Scholar
  14. Current JR, Schilling DA (1990) Analysis or errors due to demand data aggregation in the set covering and maximal covering location problems. Geogr Anal 22:166–122Google Scholar
  15. Daskin MS, Haghani AE, Khanal M, Malandraki C (1989) Aggregation effects in maximum covering models. Ann Oper Res 18:115–140CrossRefGoogle Scholar
  16. Densham P, Rushton G (1992) Strategies for solving large location–allocation problems by heuristic methods. Environ Plann A 24:289–304CrossRefGoogle Scholar
  17. Drezner T (1994) Locating a single new facility among existing, unequally attractive facilities. J Reg Sci 34:237–252CrossRefGoogle Scholar
  18. Eiselt HA, Laporte G (1989) The maximum capture problem in a weighted network. J Reg Sci 29:433–439CrossRefGoogle Scholar
  19. Emir-Farinas H, Francis RL (2005) Demand point aggregation for planar covering location problems. Ann Oper Res 136:175–192CrossRefGoogle Scholar
  20. Environmental Systems Research Institute (2006) Census 2000 TIGER/Line Data. http://www.arcdata.esri.com/data/tiger2000/tiger_download.cfm (last accessed 20 February 2006)
  21. Erkut E, Bozkaya B (1999) Analysis of aggregation errors for the p-median problem. Comput Oper Res 26:1075–1096CrossRefGoogle Scholar
  22. Fischer K (2002) Sequential discrete p-facility models for competitive location planning. Ann Oper Res 111:253–270CrossRefGoogle Scholar
  23. Fisher PF, Langford M (1996) Modeling sensitivity to accuracy in classified imagery: a study of areal interpolation by dasymetric mapping. Prof Geogr 48:299–309CrossRefGoogle Scholar
  24. Flowerdew R, Green M, Kehris E (1991) Using areal interpolation methods in geographic information. Pap Reg Sci 70:303–315CrossRefGoogle Scholar
  25. Fotheringham AS, Curtis A, Densham PJ (1995) The zone definition problem and location–allocation modeling. Geogr Anal 27:60–77CrossRefGoogle Scholar
  26. Francis RL, Lowe TJ, Tamir A (2000) Aggregation error bounds for a class of location problems. Oper Res 48:294–307CrossRefGoogle Scholar
  27. Ghosh A, Craig CS (1984) A location model for facility planning in a competitive environment. Geogr Anal 16:39–51CrossRefGoogle Scholar
  28. Goodchild MF (1979) The aggregation problem in location allocation. Geogr Anal 11:240–255CrossRefGoogle Scholar
  29. Goodchild MF, Lam N (1980) Areal interpolation: a variant of the traditional spatial problem. Geoprocessing 1:297–312Google Scholar
  30. GRANIT (2006) 2000 census block shapefiles. http://www.granit.sr.unh.edu/ (last accessed 2 February 2006)
  31. Hillsman E (1984) The p-median structure as a unified linear model for location–allocation analysis. Environ Plann A 16:305–318CrossRefGoogle Scholar
  32. Hillsman EL, Rhoda R (1978) Errors in measuring distances from populations to service centers. Ann Reg Sci Assoc 12:74–88CrossRefGoogle Scholar
  33. Hotelling H (1929) Stability in competition. Econ J 39:41–57CrossRefGoogle Scholar
  34. Kuhn HW, Kuenne RE (1962) An efficient algorithm for the numerical solution of the generalized Weber problem in spatial economics. J Reg Sci 4:21–42CrossRefGoogle Scholar
  35. Kwan MP, Murray AT, O’Kelly M, Tiefelsdorf M (2003) Recent advances in accessibility research: representation, methodology and applications. J Geogr Syst 5:129–138CrossRefGoogle Scholar
  36. Lam N (1983) Spatial interpolation methods: a review. Am Cartogr 10:129–149CrossRefGoogle Scholar
  37. Langford M, Unwin D (1994) Generating and mapping population density surfaces within a geographical information system. Cartogr J 31:21–26Google Scholar
  38. Mennis J (2003) Generating surface models of population using dasymetric mapping. Prof Geogr 55:31–42Google Scholar
  39. Miller HJ (1996) GIS and geometric representation in facility location problems. Int J Geogr Inf Syst 10:791–816CrossRefGoogle Scholar
  40. Miller HJ (2003) Representation and spatial analysis in geographic information systems. Ann Assoc Am Geogr 93:574–594CrossRefGoogle Scholar
  41. Mrozinzki RD, Cromley RG (1999) Singly- and doubly-constrained methods of areal interpolation for vector-based GIS. Trans GIS 3:285–301CrossRefGoogle Scholar
  42. Murray AT, Gottsegen JM (1997) The influence of data aggregation on the stability of p-median location model solutions. Geogr Anal 29:200–213CrossRefGoogle Scholar
  43. Murray AT, O’Kelly M (2002) Assessing representation error in point-based coverage modeling. J Geogr Syst 4:171–191CrossRefGoogle Scholar
  44. Plastria F (2001) Static competitive facility location: an overview of optimization approaches. Eur J Oper Res 129:461–470CrossRefGoogle Scholar
  45. Plastria F, Carrizosa E (2004) Optimal location and design of a competitive facility. Math Program 100:247–265CrossRefGoogle Scholar
  46. ReVelle C (1986) The maximum capture or “sphere of influence” location problem: hotelling revisited on a network. J Reg Sci 26:343–358CrossRefGoogle Scholar
  47. Serra D, ReVelle C (1994) Market capture by two competitors: the preemptive location problem. J Reg Sci 34:549–561CrossRefGoogle Scholar
  48. Serra D, Marianov V, ReVelle C (1992) The maximum-capture hierarchical location problem. Eur J Oper Res 62:363–371CrossRefGoogle Scholar
  49. Serra D, ReVelle C, Rosing K (1999) Surviving in a competitive spatial market: the threshold capture model. J Reg Sci 39:637–652CrossRefGoogle Scholar
  50. Taylor PJ (1977) Quantitative methods in geography: an introduction to spatial analysis. Houghton Mifflin, BostonGoogle Scholar
  51. Teitz M, Bart P (1968) Heuristic methods for estimating generalized vertex median of a weighted graph. Oper Res 16:955–961Google Scholar
  52. Thill JC (2000) Network competition and branch differentiation with consumer heterogeneity. Ann Reg Sci 34:451–468CrossRefGoogle Scholar
  53. Tobler WR (1979) Smooth pycnophylactic interpolation for geographical regions. J Am Stat Assoc 74:519–530CrossRefGoogle Scholar
  54. Weber A (1929) Theory of the location of industries. Translated by C.J. Friedrich. University of Chicago Press, ChicagoGoogle Scholar
  55. White M (1980) A survey of the mathematics of maps. In: Proceedings of Auto-Carto IV. American congress on surveying and mapping and the American society of photogrammetry. Falls Church, Virginia, pp 82–96Google Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Department of Geography U-4148University of ConnecticutStorrsUSA

Personalised recommendations